14,937 research outputs found
Cubic Curves, Finite Geometry and Cryptography
Some geometry on non-singular cubic curves, mainly over finite fields, is
surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are
classified accordingly. The group structure and the possible numbers of
rational points are also surveyed. A possible strengthening of the security of
elliptic curve cryptography is proposed using a `shared secret' related to the
group law. Cubic curves are also used in a new way to construct sets of points
having various combinatorial and geometric properties that are of particular
interest in finite Desarguesian planes.Comment: This is a version of our article to appear in Acta Applicandae
Mathematicae. In this version, we have corrected a sentence in the third
paragraph. The final publication is available at springerlink.com at
http://www.springerlink.com/content/xh85647871215644
Weight enumerators of Reed-Muller codes from cubic curves and their duals
Let be a finite field of characteristic not equal to or
. We compute the weight enumerators of some projective and affine
Reed-Muller codes of order over . These weight enumerators
answer enumerative questions about plane cubic curves. We apply the MacWilliams
theorem to give formulas for coefficients of the weight enumerator of the duals
of these codes. We see how traces of Hecke operators acting on spaces of cusp
forms for play a role in these formulas.Comment: 19 pages. To appear in "Arithmetic, Geometry, Cryptography, and
Coding Theory" (Y. Aubry, E. W. Howe, C. Ritzenthaler, eds.), Contemp. Math.,
201
A criterion to rule out torsion groups for elliptic curves over number fields
We present a criterion for proving that certain groups of the form do not occur as the torsion subgroup of
any elliptic curve over suitable (families of) number fields. We apply this
criterion to eliminate certain groups as torsion groups of elliptic curves over
cubic and quartic fields. We also use this criterion to give the list of all
torsion groups of elliptic curves occurring over a specific cubic field and
over a specific quartic field.Comment: 13 page
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